We study a nonlocal porous medium diffusion equation with anisotropic fractional pressure and develop a finite element method on bounded Lipschitz domains. We prove convergence to a nonnegative weak solution and present 2D experiments illustrating anisotropy, nonlocal effects, and decay to steady state.

@article{Fronzoni2026,
  author       = {Stefano Fronzoni},
  title        = {Finite element approximation of an anisotropic porous medium equation with fractional pressure},
  journal      = {arXiv preprint arXiv:2604.13014},
  year         = {2026},
}

Finite Element method for the Dirichlet problem for the integral fractional Laplacian $(-\Delta)^s$, $s\in(0,1)$, introducing bases of the form $\delta^s\times$ (piecewise linear functions). The method exploits the improved regularity of $u/\delta^s$ and achieves higher convergence rates (order $h^{2-s}$ under standard smoothness).

@article{delTesoGomezCastroFronzoni2025,
  title   = {Finite Elements with weighted bases for the fractional Laplacian},
  author  = {del Teso, F{\'e}lix and G{\'o}mez Castro, David and Fronzoni, Stefano},
  journal = {arXiv preprint arXiv:2511.01727},
  year    = {2025}
}

A fast, accurate numerical method based on Finite Elements and rational approximations for the inverse of the spectral fractional Laplacian. The method is applied to evolutionary PDEs that involve the fractional Laplacian through an interaction potential (the fractional porous medium equation and the fractional Keller-Segel equation) with numerical validation of qualitative properties.

@article{CarrilloNakatsukasaSuliFronzoni2025,
  title   = {A minimax method for the spectral fractional Laplacian and related evolution problems},
  author  = {Carrillo, Jos{\'e} A. and Nakatsukasa, Yuji and S{\"u}li, Endre and Fronzoni, Stefano},
  journal = {arXiv preprint arXiv:2505.20560},
  year    = {2025}
}

Study that quantifies the rate at which a nonlocal porous-medium approximation converges to the local model in one dimension. The analysis exploits the so-called Evolutionary Variational Inequality for both the nonlocal and local equations, as well as a priori estimates, and provides numerical evidence using a Finite Volume scheme, suggesting possible improvements.

@article{CarrilloElbarSkrzeczkowskiFronzoni2025,
  title   = {Rate of Convergence for a Nonlocal-to-local Limit in One Dimension},
  author  = {Carrillo, Jos{\'e} A. and Elbar, Charles and Skrzeczkowski, Jakub and Fronzoni, Stefano},
  journal = {Communications on Pure and Applied Analysis},
  year    = {2025},
  doi     = {10.3934/cpaa.2025114}
}

Finite Element scheme for a porous medium equation with a nonlocal pressure, given by the spectral Neumann Laplacian. The study performs a rigorous passage to the limit as the spatial and temporal discretisation parameters tend to zero and shows that a subsequence of Finite Element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem. Exponential decay of the total energy associated with the problem is also established.

@article{CarrilloSuliFronzoni2025,
  title   = {Finite Element Approximation of the Fractional Porous Medium Equation},
  author  = {Carrillo, Jos{\'e} A. and S{\"u}li, Endre and Fronzoni, Stefano},
  journal = {Numerische Mathematik},
  year    = {2025},
  doi     = {10.1007/s00211-025-01486-3}
}

Conservation-law formulation of a fractional Laplacian suitable for Finite Volume schemes, allowing direct no-flux boundary prescription and capturing anomalous diffusion. Numerical exploration of properties of the fractional heat equation and Lévy-Fokker-Plack equation with respect to their stationary states and long-time asymptotics.

@article{BailoCarrilloGomezCastroFronzoni2024,
  title   = {A finite-volume scheme for fractional diffusion on bounded domains},
  author  = {Bailo, Rafael and Carrillo, Jos{\'e} A. and G{\'o}mez Castro, David and Fronzoni, Stefano},
  journal = {European Journal of Applied Mathematics},
  year    = {2024},
  doi     = {10.1017/S0956792524000172}
}